Find the Sum of the Infinite Geometric Series 1+1/3+1/9+1/27

Problem

1+1/3+1/9+1/27+…

Solution

1. Identify the first term a and the common ratio r of the geometric series.

a=1

r=(1/3)/1=1/3

2. Check the convergence condition for an infinite geometric series, which requires |r|<1

|1/3|<1

3. Apply the formula for the sum of an infinite geometric series, S=a/(1−r)

S=1/(1−1/3)

4. Simplify the denominator by performing the subtraction.

1−1/3=2/3

5. Calculate the final sum by dividing the first term by the simplified denominator.

S=1/(2/3)=3/2

Final Answer

1+1/3+1/9+1/27+⋯=3/2

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